How can inner parts of a mesh be selected?
Select using Limit Selection to Visible and Hide

One possible way to achieve this goal is to enable Limit Selection to Visible, select the outer surface, Invert Selection, then Hide it. This method could be particularly useful if your model has lots of cavities you want to hide.

For example, this mesh has an inner cavity with a pink ball in it:  

 

To hide the model's outer shell, do this:   


Select the object, then enter Edit mode with TAB
Enable Limit Selection to Visible in the 3D Viewport Header so that only faces visible to the camera can be selected:  


Toggle into Quad View with CTRLALTQ


Switch to using the Circle Select tool (C), then increase the size of the selection circle by scrolling up with the scroll wheel (MMB) until you can select most of your model with 1 click
Now using the four windows of the Quad View, select all the outside faces, being careful not to select anything in the cavities:  



Continue selecting until all outer faces are selected.
Note: 


Since there is no "Back View", you will have to rotate in the User defined window to select those
Once you have used the Circle Select tool to select in one window, you must stop using the tool while in that window before you can begin to use it in another window. To stop using it, click the LMB (Left Mouse Button) in the window you began using it in

Rotate the mesh in the User defined window (upper right window) to see if you've successfully selected all the outer mesh. Keep selecting until it's all selected.
Note: 


Sometimes you have to zoom in and look closely in order for the Circle Selection tool to actually select the geometry
If you accidentally select something you don't want selected, you can deselect it with the MMB (Middel Mouse Button/Scroll Wheel) while still using the Circle Selection tool  

Once all the outer mesh is selected, Hide the selected faces by pressing H

  
If you find you have accidentally hidden parts of the mesh you do not want to hide, then Unhide with ALTH, edit your selection, then Hide again. Repeat as necessary:  


Once you've made sure all your cavities are properly selected, Unhide the outer mesh, then Invert Selection with CTRLI, then Hide

 

Now all the cavities are hidden.